Student APS
Speaker: Omkar Ramdas, IIT Bombay
Host: Niranjan Balachandran
Time, day and date: 11:00:00 AM - 12:00:00 PM, Wednesday, February 18
Venue: Room 105
Title: Extremal properties of permutations
Abstract: This talk will broadly be divided into two parts. In the first part, I'll define some special families of permutations and we'll see some optimal bounds on the sizes of such families. In the second part, I'll define what caged sequences are, for any given permutation and we'll see their asymptotic properties.

Student APS
Speaker: Swapan Jana, IIT Bombay
Host: Sourav Pal
Title: ANDOˆ ’S DILATION AND BOHR-THEORETIC ASPECTS OF THE MULTIVARIATE VON NEUMANN INEQUALITY FOR BANACH SPACE OPERATORS
Time, day and date: 12:00:00 PM – 1:00:00 PM, Wednesday, February 18
Venue: Room 105
Abstract: The celebrated Bela Sz.-Nagy dilation theorem states that every contraction ´ T on a Hilbert space H dilates to an isometry and the minimal isometric dilation space is uniquely determined (up to unitary equivalence) by H ⊕2 `2(DT ), which equals H ⊕2 `2(H ) if T is a strict contraction. In 1963, Ando proved that a commuting pair of contractions on a Hilbert space ˆ H dilates to a commuting pair of isometries on H ⊕2 `2(H ⊕2 H ⊕2 H ⊕2 H ). We show that a commuting pair of strict contractions on a Hilbert space H admits an isometric dilation on H ⊕2 `2(H ⊕2 H ). Moreover, we prove that H ⊕2 `2(H ⊕2 H ) is a minimal isometric dilation space, which shows that minimal isometric dilation space is independent of the choice of the commuting pair of strict contractions on the underlying Hilbert space. An immediate application of our construction is that we extend this mechanism to construct an isometric dilation of a commuting pair of strict contractions on a Banach space. This part is based on a joint work with Prof. Sourav Pal. For c ∈ C n (n ≥ 1), let Gc be the collection of tuples (T1,...,Tn) of Banach space operators satisfying kTik ≤ |ci | for 1 ≤ i ≤ n. Using a multidimensional analogue of Bohr’s theorem, we show that the closed polydisk D n (R) is a spectral set for Gc if and only if c ∈ B(D n (R)), the Bohr zone of D n (R). In contract to the one-variable case, we prove that D n (R) is not a minimal spectral set of Gc for any c ∈ B(D n (R)) when n ≥ 2. This part is based on an ongoing joint work with Prof. Sourav Pal and Dr. Himadri Halder.

Student APS
Speaker: Shivani, IIT Bombay
Host: Neela Nataraj
Title: Discontinuous Galerkin finite element methods for linear and semilinear fourth-order problems in smectic liquid crystals
Time, day and date: 2:00:00 PM – 3:00:00 PM, Wednesday, February 18
Venue: Ramanujan Hall
Abstract: In this talk, we will discuss the analysis and numerical approximation of a regular solution to a semilinear fourth-order partial differential equation defined on a polygonal domain, modelling the density variation of Smectic-A liquid crystals. First, we will discuss the discontinuous Galerkin finite element method for the corresponding linear problem. Building upon this, we will discuss the analysis of the semilinear fourth-order problem. For the nonlinear model, we establish existence, local uniqueness, and optimal order error estimates in the energy norm under the assumption that the exact solution $u\in H^{2+\alpha}(\Omega)$ with $\alpha>0$. Numerical experiments validating the theoretical results will also be presented.

Special Session
Speaker: Madhusudan Manjunath, IIT Bombay
Title: Mathematics for Well-Being
Time, day and date: 4:00:00 PM - 5:15:00 PM, Wednesday, February 18
Venue: Ramanujan Hall
Abstract: We ask fundamental questions concerning Mathematics and its potential for Well-being. Taking guidance from an ancient school of Indian philosophy, we sketch candidate answers to these questions. The talk assumes no special background in either Mathematics or Philosophy.

Student Seminar
Speaker: Om Milind Joglekar, IIT Bombay
Host: Suman Kumar Sahoo
Title: Perron-Frobenius Theory: Matrices & Graphs
Time, day and date: 5:30:00 PM – 6:30:00 PM, Wednesday, February 18
Venue: Ramanujan Hall
Abstract: This talk examines the Perron-Frobenius theorem, focusing on the existence of a strictly positive eigenvector and the spectral consequences of irreducibility. We will specifically connect these analytic properties to graph-theoretic concepts like connectivity and cycle periodicity.